3.32.20 \(\int \frac {1}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [3120]

3.32.20.1 Optimal result

Integrand size = 53, antiderivative size = 697 \[ \int \frac {1}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (1944 b c^4+3640 a b x\right ) \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (-2106 b c^3+6561 a c^7 x\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (2340 b c^2-4374 a c^6 x\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (-2730 b c+3645 a c^5 x\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\sqrt {-b+a^2 x^2} \left (3640 b \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+6561 c^7 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}-4374 c^6 \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+3645 c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{4860 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {182 b \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{243 \sqrt {3} a c^{16/3}}+\frac {182 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{729 a c^{16/3}}-\frac {91 b \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{729 a c^{16/3}} \]

output

1/4860*((1944*b*c^4+3640*a*b*x)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)+(6 
561*a*c^7*x-2106*b*c^3)*(a*x+(a^2*x^2-b)^(1/2))^(1/4)*(c+(a*x+(a^2*x^2-b)^ 
(1/2))^(1/4))^(2/3)+(-4374*a*c^6*x+2340*b*c^2)*(a*x+(a^2*x^2-b)^(1/2))^(1/ 
2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)+(3645*a*c^5*x-2730*b*c)*(a*x+(a 
^2*x^2-b)^(1/2))^(3/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)+(a^2*x^2-b) 
^(1/2)*(3640*b*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)+6561*c^7*(a*x+(a^2* 
x^2-b)^(1/2))^(1/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)-4374*c^6*(a*x+ 
(a^2*x^2-b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)+3645*c^5* 
(a*x+(a^2*x^2-b)^(1/2))^(3/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3)))/a/ 
c^5/(a*x+(a^2*x^2-b)^(1/2))^(5/4)+182/729*b*arctan(1/3*3^(1/2)+2/3*(c+(a*x 
+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)*3^(1/2)/c^(1/3))*3^(1/2)/a/c^(16/3)+182/7 
29*b*ln(-c^(1/3)+(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3))/a/c^(16/3)-91/72 
9*b*ln(c^(2/3)+c^(1/3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)+(c+(a*x+(a^ 
2*x^2-b)^(1/2))^(1/4))^(2/3))/a/c^(16/3)
 

3.32.20.2 Mathematica [A] (verified)

Time = 2.34 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\frac {3 \sqrt [3]{c} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3} \left (729 c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4} \left (9 c^2-6 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+5 \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )+2 b \left (972 c^4-1053 c^3 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+1170 c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}-1365 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}+1820 \left (a x+\sqrt {-b+a^2 x^2}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+3640 \sqrt {3} b \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+3640 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-1820 b \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{14580 a c^{16/3}} \]

input

Integrate[1/((a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^ 
2])^(1/4))^(1/3)),x]
 

output

((3*c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)*(729*c^5*(a*x + S 
qrt[-b + a^2*x^2])^(5/4)*(9*c^2 - 6*c*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + 5 
*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) + 2*b*(972*c^4 - 1053*c^3*(a*x + Sqrt[-b 
+ a^2*x^2])^(1/4) + 1170*c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]] - 1365*c*(a*x 
+ Sqrt[-b + a^2*x^2])^(3/4) + 1820*(a*x + Sqrt[-b + a^2*x^2]))))/(a*x + Sq 
rt[-b + a^2*x^2])^(5/4) + 3640*Sqrt[3]*b*ArcTan[(1 + (2*(c + (a*x + Sqrt[- 
b + a^2*x^2])^(1/4))^(1/3))/c^(1/3))/Sqrt[3]] + 3640*b*Log[-c^(1/3) + (c + 
 (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)] - 1820*b*Log[c^(2/3) + c^(1/3)*( 
c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^ 
2])^(1/4))^(2/3)])/(14580*a*c^(16/3))
 

3.32.20.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}} \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {1}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}dx\)

input

Int[1/((a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1 
/4))^(1/3)),x]
 

output

$Aborted
 

3.32.20.3.1 Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.32.20.4 Maple [F]

\[\int \frac {1}{\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}}d x\]

input

int(1/(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3 
),x)
 

output

int(1/(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3 
),x)
 

3.32.20.5 Fricas [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 1036, normalized size of antiderivative = 1.49 \[ \int \frac {1}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Too large to display} \]

input

integrate(1/(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4) 
)^(1/3),x, algorithm="fricas")
 

output

[1/14580*(5460*sqrt(1/3)*b^2*c*sqrt(-1/c^(2/3))*log(6*sqrt(1/3)*(a*c^(2/3) 
*x - sqrt(a^2*x^2 - b)*c^(2/3))*(a*x + sqrt(a^2*x^2 - b))^(3/4)*(c + (a*x 
+ sqrt(a^2*x^2 - b))^(1/4))^(2/3)*sqrt(-1/c^(2/3)) - 3*(a*c^(2/3)*x + sqrt 
(1/3)*(a*c*x - sqrt(a^2*x^2 - b)*c)*sqrt(-1/c^(2/3)) - sqrt(a^2*x^2 - b)*c 
^(2/3))*(a*x + sqrt(a^2*x^2 - b))^(3/4)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/ 
4))^(1/3) + 3*(a*c*x - sqrt(1/3)*(a*c^(4/3)*x - sqrt(a^2*x^2 - b)*c^(4/3)) 
*sqrt(-1/c^(2/3)) - sqrt(a^2*x^2 - b)*c)*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 
 2*b) - 1820*b^2*c^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3) + 
 (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c^(1/3) + c^(2/3)) + 3640*b^2 
*c^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) - c^(1/3)) + 3*(6 
561*b*c^8 - 2106*a*b*c^4*x + 2106*sqrt(a^2*x^2 - b)*b*c^4 + 8*(486*a^2*c^5 
*x^2 - 243*b*c^5 + 455*a*b*c*x - (486*a*c^5*x + 455*b*c)*sqrt(a^2*x^2 - b) 
)*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 15*(243*b*c^6 - 182*a*b*c^2*x + 182*sq 
rt(a^2*x^2 - b)*b*c^2)*sqrt(a*x + sqrt(a^2*x^2 - b)) - 18*(243*b*c^7 - 130 
*a*b*c^3*x + 130*sqrt(a^2*x^2 - b)*b*c^3)*(a*x + sqrt(a^2*x^2 - b))^(1/4)) 
*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3))/(a*b*c^6), 1/14580*(10920*sq 
rt(1/3)*b^2*c^(2/3)*arctan(sqrt(1/3) + 2*sqrt(1/3)*(c + (a*x + sqrt(a^2*x^ 
2 - b))^(1/4))^(1/3)/c^(1/3)) - 1820*b^2*c^(2/3)*log((c + (a*x + sqrt(a^2* 
x^2 - b))^(1/4))^(2/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c^(1/ 
3) + c^(2/3)) + 3640*b^2*c^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/...
 

3.32.20.6 Sympy [F]

\[ \int \frac {1}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{\sqrt [3]{c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}}\, dx \]

input

integrate(1/(a*x+(a**2*x**2-b)**(1/2))**(1/4)/(c+(a*x+(a**2*x**2-b)**(1/2) 
)**(1/4))**(1/3),x)
 

output

Integral(1/((c + (a*x + sqrt(a**2*x**2 - b))**(1/4))**(1/3)*(a*x + sqrt(a* 
*2*x**2 - b))**(1/4)), x)
 

3.32.20.7 Maxima [F]

\[ \int \frac {1}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {1}{{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}} \,d x } \]

input

integrate(1/(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4) 
)^(1/3),x, algorithm="maxima")
 

output

integrate(1/((a*x + sqrt(a^2*x^2 - b))^(1/4)*(c + (a*x + sqrt(a^2*x^2 - b) 
)^(1/4))^(1/3)), x)
 

3.32.20.8 Giac [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Timed out} \]

input

integrate(1/(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4) 
)^(1/3),x, algorithm="giac")
 

output

Timed out
 

3.32.20.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/3}} \,d x \]

input

int(1/((a*x + (a^2*x^2 - b)^(1/2))^(1/4)*(c + (a*x + (a^2*x^2 - b)^(1/2))^ 
(1/4))^(1/3)),x)
 

output

int(1/((a*x + (a^2*x^2 - b)^(1/2))^(1/4)*(c + (a*x + (a^2*x^2 - b)^(1/2))^ 
(1/4))^(1/3)), x)